On the Spherical Indicatrices of a Timelike Curve as Generalized Helices in Minkowski 3-Space

Yıl 2022, Cilt: 10 Sayı: 1, 50 - 54, 15.04.2022

Arfah Arfah , Gül Güner Tuğ

Öz

In the present paper, we investigate new properties of the spherical indicatrices of a timelike curve in Minkowski 3-space $\mathbb{E}_1^3$. We focus on the conditions of the spherical indicatrix to be a generalized helix depending on its causal character. We also give some integral equations by defining the axis of the helix in the means of the local frame.

Anahtar Kelimeler

Null curves, null helices, spherical indicatrices, Minkowski space

Kaynakça

• [1] Ali A.T., New spherical curves and their spherical indicatrices, Global journal of advance research on classical and modern geometries, 2(1) (2009), 28–38.
• [2] Arfah A., On causal characterization of spherical indicatrices of timelike curves in Minkowski 3-space, Hagia sophia journal of geometry, 2(2) (2020), 26–37.
• [3] Bilici, M., Ali, A., On the natural lift curves for the involute spherical indicatrices in Minkowski 3- space, Malaya journal of mathematics, 5 (2017), 407–415.
• [4] Chung, S.K., A study on spherical indicatrix of space curve in E3, Journal of mathematical education, 20(3) (1982), 23–26.
• [5] Do Carmo, P.D., Differential Geometry of Curves and Surfaces, Prentice Hall, New Jersey, 1976.
• [6] Erdogan M., Yilmaz G., Null generalized and slant helices in 4-dimensional Lorentz-Minkowski space, Int. J., Contemp., Math. Sci., 3(23) (2008), 1113–1120.
• [7] Formiga, J., Romero, C., On the differential geometry of curves in Minkowski space, American Journal of Physics, 74 (2006), 1012–1016. https://doi.org/10.1119/1.2232644.
• [8] Lipschutz, M.M., Schaum’s outline of differential geometry, McGraw-Hill, Canada, 1969.
• [9] Liu H., Yuan Y., Pitch functions of ruled surfaces and B-scrolls in Minkowski 3-space, Journal of geometry and physics, 62 (2012), 47–52.
• [10] Lopez, R., Differential geometry of curves and surface in Lorentz-Minkowski space, International electronic journal of geometry, 1(2) (2014), 44–107.
• [11] Lucas, A.A., Lambin, P., Diffraction by DNA, carbon nanotubes and other helical nanostructures, Rep. Prog. Phys., 68 (2005), 1181-–1249.
• [12] Mohajan, H.K., Minkowski geometry and space-time manifold in relativity,Journal of enviromental techniques, 1 (2013), 101–109.
• [13] O’neil, B., Semi-Riemannian geometry and its applications to relativity, Academic press, New York, 1983.
• [14] Philips, J., Freedom in machinery, Cambridge University Press, New York, 1990.
• [15] Qian, J., Liu, J., Tian, X., Kim, Y.H., Structure functions of pseudo null curves in Minkowski 3-space. Mathematics, 8(1) 2020, 75–84. https://doi.org/10.3390/math8010075.
• [16] Shinkin, E.V., Handbook and atlas of curves, CRC Press, Florida, 1995.
• [17] Struik, D.J., Lectures on Classical Differential Geometry, 2nd ed., Dover, Mineola, 1988.
• [18] Synge, J.L., Time-like helices in flat spacetime, Proc. Irish. Acade. A., 65 (1967), 27—42.